Here is a way to estimate your serve speed from digital video footage
of the serve using the following 3 pieces of information:

The frame rate of your camera. It is usually
29.97 frames per second in USA, unless you used some special feature
like the 240 fps "smooth slow motion" in some Sony camcorders.

Estimate of the distance traveled in the air by your serve. The distance from the middle
of the baseline to the cross of the service T is 60 feet; the distance
from the middle of the baseline to the wide coners of the
service box is 61.5 ft. The fact that the serve is hit from a height
of 8 or 9 feet increases the distance by 0.5 ft. But typically serves
are struck a little inside the baseline, so you have to reduce by that
distance. If your serve lands about a foot inside the service line, you have
to estimate and subtract that distance as well.

Number of frames. Play the video in frame by frame mode, count the number of frames
between contact with racket and subsequent contact with the floor. You may have
to estimate partial frames.

An example of an Andy Roddick serve from the 2007 AO is included for those who wish
to calculate an accurate calculation or who wish to verify the validity of the Speed Calculator.
Click Here for Example.

The calculation below relies on the reported finding that a serve with an initial speed of
mph
slows down to
mph
over a distance of
feet

Serve Speed Calculator

Frame rate of your camera

frames/sec

Distance traveled in air

feet

Number of frames

Initial Speed

mph

Final Speed

mph

Avg Speed until bounce

mph

Distance(recalc)

feet

Travel Time

sec

Fall due to gravity

ft

Formula used in the above calculation

$K = \frac{\ln(V_i/V_f)}{S}$
$V_i$ = Initial velocity of a known serve, measured in other ways, in mph
$V_f$ = Final velocity of this known serve, measured in other ways, in mph
$S$ = Distance traveled by this known serve, in feet
5280 = number of feet in one 1 mile
3600 = number of seconds in 1 hour
$e$ & $ln$ : Anyone who would bother to
read this far should know what they are

The main assumptions are :

The motion of the ball after the hit is only affected
by the drag force of the air alone, which is assumed to be
proportional to the square of the veolocity. There is the obvious
force of gravity, but being a vertical force, it doesn't significantly
affect the calculation.

A $V_0$ mph serve slows down to $V_f$ mph over S feet.
We need these numbers from some external source such as
this
website (unfortunately, it no longer works) reporting a study done on Sampras' serve

The full derivation

The model for motion under a drag force is

$ \frac{dv}{dt} = \frac{-kv^2}{m}$

Note that this is a statement of Newton's second law,
acceleration = Force/mass.
Drag force is $-kv^2$ where $k$ is
a constant related to the drag coefficient.
Let us make up a modified drag coefficient $c = k/m$ to simplify our job
since we only care about one mass, that of a tennis ball

$ \frac{dv}{dt} = \frac{-c}{m}$

The solution to the above separable first order differential equation is

$ v = \frac{1}{ct + D} $

D is the constant of integration, and when you set t=0 above, you can see that
D must be equal to $\frac{1}{v_0}$

Therefore

$v = \frac{1}{ct + \frac{1}{v_0}}$ .....(1)

where v0 is the initial velocity, the velocity immediately
after contact with racquet.

Integrating the solution for v, we get distance s

$s = \frac{\ln(ct + \frac{1}{v_0})}{c} + E$

E is yet another constant of integration.
Knowing that the distance s is 0 when time t=0,
we calculate $E = -\ln(1/v_0)/c$

Substituting for E,

$s = \frac{\ln(ct + 1/v_0)}{c} - \frac{\ln(1/v_0)}{c}$

$s = \frac{\ln(ct + 1/v_0) - \ln(1/v_0)}{c}$

$ s = \frac{\ln(ctv_0 + 1)}{c} $ ...... (2)

$ e^{sc} = ctv_0 + 1$

$v_0 = e^{sc}/ct - 1$ ...... (3)

This $v_0$, the initial velocity immediately after racquet impact, is
what we are after. From our camcorder
experiment, we know s, the distance traveled and we know t, the time
it took to travel that distance; t = (frames * FrameRate).
The only unknown in the above formula is c, the modified drag
coefficient for a Tennis ball. Time to calibrate our model
using a known fact about Sampras' serve - that a 120 mph serve slows
down to 87 mph over 60 feet.
Substituting the numbers into Equation (1),

We want to be able to express the formula in terms of number of video
frames rather than time.
t = (F / frame_rate_sec) / 3600
The 3600 is to convert time to hours. If distance s is expressed in feet want to convert the
distance s to miles by dividing by 5280. Thus we finally arrive at the formula:

$v_0 = \frac{e^{28.3 s/5280} - 1}{28.3 F} (FrameRate \times 3600) $

where
$v_0$ is the initial speed of the serve,
s is the distance traveled by the serve during the observed interval
F is the number frames of video it took the ball to travel this distance
frame_rate is usually 29.97 with most camcorders.

Comments about this method compared to using a consumer Radar gun

Compared to using a radar gun, this method requires several extra steps.

There are several sources of error, such as estimating the number of
frames and the actual distance, but they are all manageable and in
your control( you can minimize them by being careful).
A consumer Radar can give significantly wrong speeds
depending on how far the ball traveled before entering the range of
the radar and the angle between direction of the ball and the Radar's
line of sight at the instant it "sees" the ball. Also, once in a while,
Radars give you totally wild numbers. If you suddenly hit one serve
that is 5mph mpre than every other serve, you will never know if it
was an erroneous reading or if you actually did something very right.
That will never happen with
Video method. If your serve took 0.5 secs to bounce, it is going to be
15 frames in the video. If you serve 5 mph faster, it will be a whole frame
quicker and impossible to miss. Even better, you can replay the video
and study what you did differently and try to incorporate that
permanently into your service motion.

This method gives you very visual proof that anyone can see. They
don't have to take your word.

You can tape someone while they are playing a match and estimate their
speed. You can walk around your club with a camcorder and measure the
pace of anyone's serve or groundstrokes.
OTOH, A consumer radar can never be used in a match because it need to
be placed on the court.